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Linear Functions & Graphing from Standard Form & Graphing Using the Intercepts Algebra 1 Glencoe McGraw-Hill JoAnn Evans A linear equation is the equation of a line. Linear equations can be written in the form of: This is known as the Standard Form of a linear equation. In Standard Form: 1. A ≥ 0 A must be positive! 2. A and B can’t both be 0…and they must both be on the same side! 3. A, B, and C must be integers with no 3x + 4y = 7 A=3 B=4 C=7 x+ – -2y = 5 A = 1 B = -2 C = 5 8x + 9y = 1 A=8 B=9 C=1 5x + –- y = 2 A = 5 B = -1 C = 2 A is the coefficient of the x term and B is the coefficient of the y term. C is a constant. Identify the values of A, B, and C in the equations on the left. Explain why the following equations are NOT considered to be in standard form. 2x 12 3y x and y need to be on the left side of equal sign y 3x 8 x term comes before the y term 2 xy7 3 “A” is not an integer 2x 4 y 6 2x 4 y 8 “A” is not greater than or equal to 1 GCF of A, B, and C is not 1 Determine whether the equation is a linear equation. If it is, write it in Standard Form. y 5 2x 2x 2x 2x y 5 The power of the variables is one. It is linear. Move the x term to the left side . 5x 3y z 2 2xy 5y 6 There are 3 variables. It is not linear. The term 2xy has two variables. It is not linear. 3 x y8 4 3 4 x 4( y) 4(8) 4 3x 4 y 32 4y 4y 3x 4 y 32 The power of the variables is one. It is linear. It can be written in Standard Form after the fraction is cleared. 5x y 1 y y 5x y 1 1( 5x) 1( y) 1(1) 5x y 1 The power of the variables is one. It is linear. Remember that A must be greater than or equal to 1 to be in Standard Form. Efficiency in Graphing……… In the last lesson, graphs were sketched by creating a table of values, plotting the corresponding points, and drawing the graph through those points. However, this is only one of several methods and is actually the least efficient of all. Consider this: * The graph of any linear equation is a line. * Two points are all that are needed to determine a line. Two convenient points to use would be the points where the line intersects the x-axis and the y-axis. y Name the x-intercepts and the y-intercepts for the green and blue lines. x The x-intercept is 3. It occurs at the point (3, 0). The x-intercept is 1, and occurs at the point (1, 0). The y-intercept is - 4. It occurs at the point (0, - 4). The y-intercept is 4, and occurs at the point (0, 4). The x-intercept of a line occurs when y = 0 because every point on the x-axis has a y-coordinate of zero. Examples: (- 4, 0) (-1, 0) (1, 0) (3, 0) y x To find the x-intercept of a line, let y = 0 and solve the equation for x. The y-intercept of a line occurs when x = 0 because every point on the y-axis has an x-coordinate of zero. Examples: (0, 4) (0, 2) (0, -1) (0, -5) y x To find the y-intercept of a line, let x = 0 and solve for y in the equation. Find the x-intercept for the equation x + 5y = 10. To find the x-intercept, LET y = 0. x + 5y = 10 x + 5(0) = 10 x = 10 original equation substitute 0 for y. solve for x The x-intercept is 10. What are the coordinates of the point on the x-axis where x = 10? (10, 0) Find the y-intercept for the equation x + 5y = 10. To find the y- intercept, LET x = 0. x + 5y = 10 (0) + 5y = 10 y = 2 original equation substitute 0 for x. solve for y The y-intercept is 2. What are the coordinates of the point on the y-axis where y = 2? (0, 2) Graph x + 5y = 10 using the intercepts. The x-intercept is 10. The y-intercept is 2. Two points determine a line. If you know the x-intercept and the y-intercept, the line can be quickly graphed. Connect the points to see the graph of x + 5y = 10. Use the intercepts of a line to sketch a Quick Graph. 5x + 3y = 15 First, find the x- and y-intercepts. x-intercept (Let y = 0) 5x + 3y = 15 5x + 3(0) = 15 5x = 15 x=3 y-intercept (Let x =0) 5x + 3y = 15 5(0) + 3y = 15 3y = 15 y=5 The x-intercept is 3. The y-intercept is 5. y x Find the x- and y-intercept of each equation on your own: 2x + 3y = 12 x-intercept: 2x + 3(0) = 12 x = 6 y-intercept: 2(0) + 3y = 12 y=4 x-intercept: 6 y-intercept: 4 y x -4x + 2y = 10 x-intercept: -4x + 2(0) = 10 -4x = 10 x = -2.5 y-intercept: -4(0) + 2y = 10 y=5 x-intercept: -2.5 y-intercept: 5 y x x - 4y = 2 x-intercept: x - 4(0) = 2 x=2 y-intercept: (0) - 4y = 2 y = -0.5 x-intercept: 2 y-intercept: -0.5 y x To find the x-intercept, let y equal zero . To find the y–intercept, let x equal zero .